Problem: What is the largest multiple of $9$ whose negation is greater than $-100$?
Solution: Dividing $-100$ by $9$ gives $-11$ with a remainder of $-1$. In other words, $$-100 = -11 \cdot 9 + (-1).$$This means that $-11 \cdot 9 = -99$ is greater than $-100$. Because the negation of $99$ is $-99$, $\boxed{99}$ is the largest multiple of $9$ whose negation is greater than $-100$.

We could also divide $-100$ by $9$ to get $-12$ with a positive remainder of $11$. However, $-12 \cdot 9$ is less than $-100$, so $$-100 = -12 \cdot 9 + 11$$would not help.